key31fa02 - COT3100.01, Fall 2002 Solution Keys to Final...

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COT3100.01, Fall 2002 Solution Keys to Final Exam (Test #3) 12/03/2002 S. Lang (12/03/2002) Part I. (36 pts., 3 pts. each) True/False, or short-answer, questions. (No explanation needed and no partial credit given.) 1. Let A = {1, 2, 3, 4} and let R A × A denote a binary relation depicted by the directed graph given in the figure. Answer each of the following True/False questions: 2. Define two real-valued functions f : [0, ) R and g : R R by the following formulas, where the set [0, ) denotes the set of non-negative real numbers, and R denotes the set of all real numbers, f ( x ) = x - 1, and g ( x ) = 2 x + 1. Answer the following questions: (a) Is function f an injection? True , because if f ( x ) = f ( y ), then x - 1 = y - 1. Adding 1 to both sides and squaring yields x = y , which proves the injection property. (b) Is the inverse of g given by the formula g - 1 ( y ) = 1 2 - y ? False , to find the inverse of function g , let y = g ( x ) = 2 x + 1, then solve for x in terms of y. Thus, x = 2 1 - y , which means g - 1 ( y ) = 2 1 - y . (c) Give the value of g ο f (4). Answer : g ο f (4) = g ( f (4)) = g ( ) 1 4 - = g (1) = 2 + 1 = 3. (d) Give (an exact description of) the range of f . Answer : [ - 1, ) = the set of real numbers that are - 1. 3. Let a and b denote positive integers. Answer the following questions: (a) If an integer c | a and c | b , then c | GCD( a , b ). True , because GCD( a , b ) = at + bu for some integers t and u , and c | ( at + bu ) if c | a and c | b are true as given in the assumption. (b) If both a and b are prime numbers, then GCD( a , b ) = 1. False , for example, GCD(2, 2) = 2. 1 2 4 3 (a) Is R anti-symmetric? True , because using the definition, whenever an edge ( a , b ) R and a b , then the edge ( b , a ) R is true. (b) Is R transitive? False , because both (2, 4) R and (4, 3) R but (2, 3) R . (c) Does the pair (1, 2) R ο R ? True , because (1, 2) is the composition of (1, 1) and (1, 2) both of which R . (d)
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key31fa02 - COT3100.01, Fall 2002 Solution Keys to Final...

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